Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1680,2,Mod(1039,1680)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1680, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 0, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1680.1039");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1680.ea (of order \(6\), degree \(2\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(13.4148675396\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 80029143512 \nu^{15} + 385788744870 \nu^{14} - 820783926284 \nu^{13} + 848040618120 \nu^{12} + \cdots + 33432180594 ) / 3707507912227 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 115452774644 \nu^{15} - 886173093256 \nu^{14} + 3342656190846 \nu^{13} + \cdots + 1622048373702 ) / 3707507912227 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 229876192869 \nu^{15} - 1434169055319 \nu^{14} + 4481995762636 \nu^{13} + \cdots + 7097366139528 ) / 3707507912227 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 234407101203 \nu^{15} + 1463062841541 \nu^{14} - 4568245377402 \nu^{13} + \cdots + 5702240414038 ) / 3707507912227 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 394538957168 \nu^{15} - 2231811901383 \nu^{14} + 6199106801734 \nu^{13} + \cdots + 1505816495286 ) / 3707507912227 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 652538131849 \nu^{15} - 3933384833664 \nu^{14} + 11921970963777 \nu^{13} + \cdots + 15430211164083 ) / 3707507912227 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 924551808442 \nu^{15} + 5807708679413 \nu^{14} - 18298804575933 \nu^{13} + \cdots - 6493327367967 ) / 3707507912227 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 139065217255 \nu^{15} + 863331553752 \nu^{14} - 2689115514869 \nu^{13} + \cdots - 665511369177 ) / 529643987461 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 1164639238978 \nu^{15} + 6965074914023 \nu^{14} - 20761156354785 \nu^{13} + \cdots - 6393030826185 ) / 3707507912227 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 785074438 \nu^{15} + 4907305158 \nu^{14} - 15343416116 \nu^{13} + 31997236762 \nu^{12} + \cdots - 3343669588 ) / 1973128213 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} + \cdots - 10233974547 ) / 3810388399 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 303278102799 \nu^{15} + 1888448618289 \nu^{14} - 5892807192287 \nu^{13} + \cdots - 2299757714928 ) / 529643987461 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 2432602484212 \nu^{15} - 15210362430228 \nu^{14} + 47566230428388 \nu^{13} + \cdots + 3660982645558 ) / 3707507912227 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 2667083154356 \nu^{15} - 16670596841871 \nu^{14} + 52123769377554 \nu^{13} + \cdots + 8941171414259 ) / 3707507912227 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 29169335295 \nu^{15} - 179860916164 \nu^{14} + 554518308281 \nu^{13} - 1139653845381 \nu^{12} + \cdots + 116597819081 ) / 22200646181 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{14} + \beta_{13} + \beta_{9} - \beta_{7} + \beta_{5} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} - \beta_{7} - 3\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{10} + 3 \beta_{9} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2 \beta_{15} + \beta_{14} + 7 \beta_{13} + \beta_{12} + 9 \beta_{10} - 2 \beta_{9} + \beta_{8} + \cdots + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 16 \beta_{15} + 5 \beta_{14} + 27 \beta_{13} - 4 \beta_{11} + 30 \beta_{10} - 16 \beta_{9} + \cdots + 23 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 8 \beta_{15} + 8 \beta_{14} + 8 \beta_{12} - 5 \beta_{11} + 8 \beta_{10} - \beta_{9} + 16 \beta_{8} + \cdots - 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 45 \beta_{14} - 47 \beta_{13} + 102 \beta_{12} - 18 \beta_{11} - 20 \beta_{10} + 47 \beta_{9} + \cdots - 127 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 50 \beta_{15} + 88 \beta_{14} + 60 \beta_{13} + 100 \beta_{12} + 95 \beta_{10} - 50 \beta_{9} + \cdots - 88 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 596 \beta_{15} + 576 \beta_{14} + 576 \beta_{13} - 120 \beta_{11} + 912 \beta_{10} - 799 \beta_{9} + \cdots - 120 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 576 \beta_{15} + 576 \beta_{14} - 288 \beta_{12} - 353 \beta_{11} + 576 \beta_{10} - 739 \beta_{9} + \cdots - 576 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 1125 \beta_{14} - 4331 \beta_{13} - 1432 \beta_{11} - 2490 \beta_{10} + 140 \beta_{9} - 3206 \beta_{7} + \cdots - 6105 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( 1603 \beta_{15} + 339 \beta_{14} - 5148 \beta_{13} + 1603 \beta_{12} - 4809 \beta_{10} + 1603 \beta_{9} + \cdots - 7468 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 5009 \beta_{14} - 5865 \beta_{13} + 4062 \beta_{11} - 4918 \beta_{10} - 5865 \beta_{9} - 18420 \beta_{8} + \cdots - 18511 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 8782 \beta_{15} + 8782 \beta_{14} - 17564 \beta_{12} + 8782 \beta_{10} - 18939 \beta_{9} + \cdots - 8782 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 4918 \beta_{14} - 95488 \beta_{13} - 100406 \beta_{12} - 22482 \beta_{11} - 45606 \beta_{10} + \cdots - 77924 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(337\) | \(421\) | \(1121\) | \(1471\) |
| \(\chi(n)\) | \(1 + \beta_{10}\) | \(-1\) | \(1\) | \(1\) | \(-1\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1039.1 |
|
0 | −0.866025 | − | 0.500000i | 0 | −0.893868 | + | 2.04963i | 0 | −2.24014 | − | 1.40775i | 0 | 0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1039.2 | 0 | −0.866025 | − | 0.500000i | 0 | −0.759216 | − | 2.10323i | 0 | 0.649221 | − | 2.56486i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1039.3 | 0 | −0.866025 | − | 0.500000i | 0 | 1.85468 | + | 1.24906i | 0 | 2.11465 | + | 1.59005i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1039.4 | 0 | −0.866025 | − | 0.500000i | 0 | 2.16443 | − | 0.561484i | 0 | −2.25578 | + | 1.38256i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1039.5 | 0 | 0.866025 | + | 0.500000i | 0 | −2.22197 | − | 0.250705i | 0 | 2.24014 | + | 1.40775i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1039.6 | 0 | 0.866025 | + | 0.500000i | 0 | −0.154373 | − | 2.23073i | 0 | −2.11465 | − | 1.59005i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1039.7 | 0 | 0.866025 | + | 0.500000i | 0 | 1.44185 | + | 1.70912i | 0 | −0.649221 | + | 2.56486i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1039.8 | 0 | 0.866025 | + | 0.500000i | 0 | 1.56847 | − | 1.59371i | 0 | 2.25578 | − | 1.38256i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1279.1 | 0 | −0.866025 | + | 0.500000i | 0 | −0.893868 | − | 2.04963i | 0 | −2.24014 | + | 1.40775i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1279.2 | 0 | −0.866025 | + | 0.500000i | 0 | −0.759216 | + | 2.10323i | 0 | 0.649221 | + | 2.56486i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1279.3 | 0 | −0.866025 | + | 0.500000i | 0 | 1.85468 | − | 1.24906i | 0 | 2.11465 | − | 1.59005i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1279.4 | 0 | −0.866025 | + | 0.500000i | 0 | 2.16443 | + | 0.561484i | 0 | −2.25578 | − | 1.38256i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1279.5 | 0 | 0.866025 | − | 0.500000i | 0 | −2.22197 | + | 0.250705i | 0 | 2.24014 | − | 1.40775i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1279.6 | 0 | 0.866025 | − | 0.500000i | 0 | −0.154373 | + | 2.23073i | 0 | −2.11465 | + | 1.59005i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1279.7 | 0 | 0.866025 | − | 0.500000i | 0 | 1.44185 | − | 1.70912i | 0 | −0.649221 | − | 2.56486i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1279.8 | 0 | 0.866025 | − | 0.500000i | 0 | 1.56847 | + | 1.59371i | 0 | 2.25578 | + | 1.38256i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 28.f | even | 6 | 1 | inner |
| 140.s | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1680.2.ea.b | yes | 16 |
| 4.b | odd | 2 | 1 | 1680.2.ea.a | ✓ | 16 | |
| 5.b | even | 2 | 1 | inner | 1680.2.ea.b | yes | 16 |
| 7.d | odd | 6 | 1 | 1680.2.ea.a | ✓ | 16 | |
| 20.d | odd | 2 | 1 | 1680.2.ea.a | ✓ | 16 | |
| 28.f | even | 6 | 1 | inner | 1680.2.ea.b | yes | 16 |
| 35.i | odd | 6 | 1 | 1680.2.ea.a | ✓ | 16 | |
| 140.s | even | 6 | 1 | inner | 1680.2.ea.b | yes | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1680.2.ea.a | ✓ | 16 | 4.b | odd | 2 | 1 | |
| 1680.2.ea.a | ✓ | 16 | 7.d | odd | 6 | 1 | |
| 1680.2.ea.a | ✓ | 16 | 20.d | odd | 2 | 1 | |
| 1680.2.ea.a | ✓ | 16 | 35.i | odd | 6 | 1 | |
| 1680.2.ea.b | yes | 16 | 1.a | even | 1 | 1 | trivial |
| 1680.2.ea.b | yes | 16 | 5.b | even | 2 | 1 | inner |
| 1680.2.ea.b | yes | 16 | 28.f | even | 6 | 1 | inner |
| 1680.2.ea.b | yes | 16 | 140.s | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{8} - 24T_{11}^{7} + 254T_{11}^{6} - 1488T_{11}^{5} + 5058T_{11}^{4} - 9300T_{11}^{3} + 6632T_{11}^{2} + 2100T_{11} + 196 \)
acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( (T^{4} - T^{2} + 1)^{4} \)
|
| $5$ |
\( T^{16} - 6 T^{15} + \cdots + 390625 \)
|
| $7$ |
\( T^{16} - 4 T^{14} + \cdots + 5764801 \)
|
| $11$ |
\( (T^{8} - 24 T^{7} + \cdots + 196)^{2} \)
|
| $13$ |
\( (T^{8} - 84 T^{6} + \cdots + 99225)^{2} \)
|
| $17$ |
\( T^{16} + \cdots + 71997768976 \)
|
| $19$ |
\( (T^{8} + 6 T^{7} + \cdots + 3969)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 3662186256 \)
|
| $29$ |
\( (T^{4} - 6 T^{3} - 30 T^{2} + \cdots + 90)^{4} \)
|
| $31$ |
\( (T^{8} + 6 T^{7} + \cdots + 21609)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 5554571841 \)
|
| $41$ |
\( (T^{8} + 156 T^{6} + \cdots + 1764)^{2} \)
|
| $43$ |
\( (T^{8} - 168 T^{6} + \cdots + 5625)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 64524128256 \)
|
| $53$ |
\( T^{16} + \cdots + 3262849744896 \)
|
| $59$ |
\( (T^{8} + 18 T^{7} + \cdots + 777924)^{2} \)
|
| $61$ |
\( (T^{8} - 132 T^{6} + \cdots + 144)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 68510864482641 \)
|
| $71$ |
\( (T^{8} + 416 T^{6} + \cdots + 10278436)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 37822859361 \)
|
| $79$ |
\( (T^{8} - 42 T^{7} + \cdots + 3969)^{2} \)
|
| $83$ |
\( (T^{8} + 180 T^{6} + \cdots + 1542564)^{2} \)
|
| $89$ |
\( (T^{8} + 12 T^{7} + \cdots + 8100)^{2} \)
|
| $97$ |
\( (T^{8} - 660 T^{6} + \cdots + 77792400)^{2} \)
|
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